Fermion Mass Hierarchies, Large Lepton Mixing and Residual Modular Symmetries

TL;DR

This paper explores how residual modular symmetries in modular-invariant flavor models can naturally generate fermion mass hierarchies without flavons, focusing on specific finite modular groups and their representations.

Contribution

It systematically analyzes fermion representation choices under residual symmetries in modular groups, proposing models that produce realistic fermion mass hierarchies and mixing without fine-tuning.

Findings

Hierarchies depend on representation decomposition under residual symmetry.

Identified specific representation pairs yielding viable mass hierarchies.

Constructed lepton models with natural mass hierarchies and mixing.

Abstract

In modular-invariant models of flavour, hierarchical fermion mass matrices may arise solely due to the proximity of the modulus $\tau$ to a point of residual symmetry. This mechanism does not require flavon fields, and modular weights are not analogous to Froggatt-Nielsen charges. Instead, we show that hierarchies depend on the decomposition of field representations under the residual symmetry group. We systematically go through the possible fermion field representation choices which may yield hierarchical structures in the vicinity of symmetric points, for the four smallest finite modular groups, isomorphic to $S_3$, $A_4$, $S_4$, and $A_5$, as well as for their double covers. We find a restricted set of pairs of representations for which the discussed mechanism may produce viable fermion (charged-lepton and quark) mass hierarchies. We present two lepton flavour models in which the charged-lepton mass hierarchies are naturally obtained, while lepton mixing is somewhat fine-tuned. After formulating the conditions for obtaining a viable lepton mixing matrix in the symmetric limit, we construct a model in which both the charged-lepton and neutrino sectors are free from fine-tuning.